General analytical framework for cooperative sensing and access trade-off optimization

In this paper, we investigate the joint cooperative spectrum sensing and access design problem for multi-channel cognitive radio networks. A general heterogeneous setting is considered where the probabilities that different channels are available, SNRs of the signals received at secondary users (SUs) due to transmissions from primary users (PUs) for different users and channels can be different. We assume a cooperative sensing strategy with a general a-out-of-b aggregation rule and design a synchronized MAC protocol so that SUs can exploit available channels. We analyze the sensing performance and the throughput achieved by the joint sensing and access design. Based on this analysis, we develop algorithms to find optimal parameters for the sensing and access protocols and to determine channel assignment for SUs to maximize the system throughput. Finally, numerical results are presented to verify the effectiveness of our design and demonstrate the relative performance of our proposed algorithms and the optimal ones.

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Fig. 1. Network model (PU: primary user, SU: secondary user)
to pair i of SUs simply as SU i. We assume that each SU
can exploit multiple available channels for transmission (e.g.,
by using OFDM technology). We will design a synchronized
MAC protocol for channel access. We assume that each chan-
nel is either in the idle or busy state for each predetermined
periodic interval, which is called a cycle in this paper.
We further assume that each pair of SUs can overhear trans-
missions from other pairs of SUs (i.e., collocated networks).
There are M primary users (PUs) each of which may or may
not use one corresponding channel for its data transmission
in any cycle. In addition, it is assumed that transmission from
any pair of SUs on a particular channel will affect the primary
receiver which receives data on that channel. The network
setting under investigation is shown in Fig. 1.
B. Cooperative Spectrum Sensing
We assume that each SU i is assigned in advance a set
of channels Si where it senses all channels in this assigned
list at beginning of each cycle. Optimization of such channel
assignment will be considered in the next section. Upon
completing the channel sensing, each SU i sends the idle/busy
states of all channels in Si to the access point (AP) for further
processing. The AP upon collecting sensing results from all
SUs will decide idle/busy status for all channels. Then, the AP
broadcasts the list of available channels to all SUs. SUs are
assumed to rely on a distributed MAC protocol to perform
access resolution where the winning SU transmits data by
using all available channels. Detailed MAC protocol design
will be elaborated later.
Let H0 and H1 denote the events that a particular PU is
idle and active, respectively (i.e., the corresponding channel
is available and busy, respectively) in a cycle. In addition,
let Pj (H0) and Pj (H1) = 1 − Pj (H0) be the probabil-
ities that channel j is available and not available for all
SUs, respectively. We assume that SUs employ an energy
detection scheme and let fs be the sampling frequency used
in the sensing period for all SUs. There are two important
performance measures, which are used to quantify the sensing
performance, namely detection and false alarm probabilities.
In particular, detection event occurs when a SU successfully
senses a busy channel and false alarm represents the situation
when a spectrum sensor returns a busy status for an idle
channel (i.e., a transmission opportunity is overlooked).
Assume that transmission signals from PUs are complex-
valued PSK signals while the noise at the SUs is indepen-
dent and identically distributed circularly symmetric complex
Gaussian CN (0, N0) [4]. Then, the detection and false alarm
probabilities for the channel j at SU i can be calculated as [4]
Pij
d εij
, τij
= Q
εij
N0
− γij
− 1
τijfs
2γij + 1
, (1)
Pij
f εij
, τij
= Q
εij
N0
− 1 τijfs
= Q 2γij + 1Q−1
Pij
d εij
, τij
+ τijfsγij
, (2)
where i ∈ [1, N] is the index of a SU link, j ∈ [1, M] is
the index of a channel, εij
is the detection threshold for an
energy detector, γij
is the signal-to-noise ratio (SNR) of the
PU’s signal at the SU, fs is the sampling frequency, N0 is the
noise power, τij
is the sensing interval of SU i on channel j,
and Q (.) is deﬁned as Q (x) = 1/
√
2π
∞
x
exp −t2
/2 dt.
We assume that a general cooperative sensing scheme,
namely a-out-of-b rule, is employed by the AP to determine
the idle/busy status of each channel based on reported sensing
results from all SUs. Under this scheme, the AP will declare
that a channel is available if a or more SUs out of b SUs report
that the underlying channel is available. The a-out-of-b rule
covers different other rules including OR, AND and Majority
rules as special cases. In particular, when a = 1, it is OR
rule; when a = b, it is AND rule; and when a = b/2 , it is
Majority rule. Let consider channel j. Let SU
j denote the set
of SUs that sense channel j and bj = SU
j be the number of
SUs sensing channel j. Then the AP’s decision on the status of
channel j will result in detection and false alarm probabilities
for this channel, which can be calculated as, respectively [8]
Pj
d εj
, τj
, aj =
bj
l=aj
Cl
bj
k=1 i1∈Φk
l
Pi1j
d
i2∈SU
j Φk
l
¯Pi2j
d (3)
Pj
f εj
, τj
, aj =
bj
l=aj
Cl
bj
k=1 i1∈Φk
l
Pi1j
f
i2∈SU
j Φk
l
¯Pi2j
f (4)
where Φk
l in (3) and (4) are particular sets with l SUs whose
sensing outcomes indicate that channel j is busy given that
this channel is indeed busy and idle, respectively; εj
= εij
,
τj
= τij
, i ∈ SU
j . For brevity, Pj
d εj
, τj
, aj and
Pj
f εj
, τj
, aj are written as Pj
d and Pj
f in the following.
C. MAC Protocol Design
We assume that time is divided into ﬁxed-size cycles and
it is assumed that SUs can perfectly synchronize with each
other (i.e., there is no synchronization error) [12]. We propose
a synchronized multi-channel MAC protocol for dynamic
spectrum sharing as follows. The MAC protocol has three
phases in each cycle utilizing one control channel, which is
assumed to be always available, as illustrated in Fig. 2. In
the ﬁrst phase, namely the sensing phase of length τ, all SUs
simultaneously perform spectrum sensing on their assigned
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Fig. 2. Timing diagram of the proposed multi-channel MAC protocol
channels. Here, we have τ = maxi τi, where τi = j∈Si
τij
is total sensing time of SU i, τij is the sensing time of SU i
on channel j, and Si is the set of channels assigned for SU
i. All SUs exchange beacon signals on the control channel to
achieve synchronization in the second phase. Then, each SU
reports its sensing results to the AP on the control channel.
The AP collects sensing results from all SUs; decide idle/status
for all channels; and broadcast this information to all SUs on
the control channel.
In the third phase, SUs participate in the contention and
the winning SU will transmit data on all vacant channels. We
assume that the length of each cycle is sufﬁciently large so that
SUs can transmit several packets during the data transmission
phase. During the data transmission phase, we assume that ac-
tive SUs employ a standard contention technique to capture the
channel similar to that in the CSMA/CA protocol. Exponential
backoff with minimum contention window W and maximum
backoff stage m0 [13] is employed in the contention phase.
For brevity, we refer to W simply as contention window in
the following. Speciﬁcally, suppose that the current backoff
stage of a particular SU is i then it starts the contention
by choosing a random backoff time uniformly distributed in
the range [0, 2i
W − 1], 0 ≤ i ≤ m0. This user then starts
decrementing its backoff time counter while carrier sensing
transmissions from other SUs on vacant channels.
Let σ denote a mini-slot interval, each of which corre-
sponds one unit of the backoff time counter. Upon hearing a
transmission from any SU, each SU will “freeze” its backoff
time counter and reactivate when the channel is sensed idle
again. Otherwise, if the backoff time counter reaches zero, the
underlying SU wins the contention. Here, two-way handshake
will be employed to transmit one data packet on the available
channel. After sending the data packet the transmitter expects
an acknowledgment (ACK) from the receiver to indicate a
successful reception of the packet. Standard small intervals,
namely DIFS and SIFS, are used before backoff time decre-
ments and ACK packet transmission as described in [13].
III. PERFORMANCE ANALYSIS, DESIGN, AND
OPTIMIZATION
A. Throughput Analysis
We assume that all SUs transmit data packets of the same
length. Let E denote the average number of vacant channels
that are correctly detected by the AP. Suppose T (τ, W) denote
the throughput achieved by all N SUs on an imaginary single-
channel network where the channel is always available. Then,
the normalized throughput per one channel achieved by our
MAC protocol can be calculated as
NT = T (τ, W)
1
M
E (5)
Here, E can be calculated as follows:
E =
M
m=1
Cm
M
i=1 j1∈Ψi
m
Pj1
(H0)
j2∈SΨi
m
Pj2
(H1) (6)
×
m
n=1
Cn
m
i1=1
n
j3∈Θ
i1
n
¯Pj3
f
j4∈Ψi
mΘ
i1
n
Pj4
f (7)
where S is the set of all M channels. The quantity (6)
represents the probability that there are m available channels,
which may or may not be correctly detected by SUs and the
AP. Here, Ψi
m denotes a particular set of m available channels
whose index is i. The second quantity (7) describes the product
of n and the probability that there are n available channels
according to the sensing decision of the AP (so the remaining
available channels are overlooked due to sensing errors) where
Θi1
n denotes the i1-th set with n available channels.
In the following, we describe how to calculate T (τ, W) by
using the technique developed by Bianchi in [13]. In particular,
we approximately assume a ﬁxed transmission probability φ
in a generic slot time. Bianchi shows that this transmission
probability can be computed from the following two equations
[13]
φ =
2 (1 − 2p)
(1 − 2p) (W + 1) + Wp (1 − (2p)
m0
)
, (8)
p = 1 − (1 − φ)
n−1
, (9)
where m0 is the maximum backoff stage, p is the conditional
collision probability (i.e., the probability that a collision is
observed when a data packet is transmitted on the channel).
For our system, there are N SUs participating in contention in
the third phase, the probability that at least one SU transmits
its data packet can be written as
Pt = 1 − (1 − φ)
N
. (10)
However, the probability that a transmission occurring on the
channel is successful given there is at least one SU transmitting
can be written as
Ps =
Nφ (1 − φ)
N−1
Pt
. (11)
The average duration of a generic slot time can be calculated
as
¯Tsd = (1 − Pt) Te + PtPsTs + Pt (1 − Ps) Tc, (12)
where Te = σ, Ts and Tc represent the duration of an empty
slot, the average time the channel is sensed busy due to a
successful transmission, and the average time the channel is
sensed busy due to a collision, respectively. These quantities
can be calculated under the basic access mechanism as [13]
⎧
⎪⎨
⎪⎩
Ts =T1
s =H+PS+SIFS+2PD+ACK+DIFS
Tc =T1
c =H+PS+DIFS+PD
H =HPHY +HMAC
, (13)
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where HPHY and HMAC are the packet headers for physical and
MAC layers, PS is the packet size in transmission time, which
is assumed to be ﬁxed in this paper, PD is the propagation
delay, SIFS is the length of a short interframe space, DIFS
is the length of a distributed interframe space, ACK is the
length of an acknowledgment. Recall that these parameters
are measured in units of bits or μs due to bit rate = 1 Mbps.
Based on these quantities, we have
T (τ, W) =
T − τ − TR
¯Tsd
PsPtPS
T
, (14)
where TR = Ntr+tb, tr is the report time from each SU to the
AP, tb the broadcast time from the AP to all SUs. Recall that
τ = maxi τi
is the total the sensing time. . denotes the ﬂoor
function and recall that T is the duration of a cycle. Note that
T −τ−TR
¯Tsd
denotes the average number of generic slot times
in one particular cycle excluding the sensing and reporting
phase. Here, we omit the length of the synchronization phase,
which is assumed to be negligible.
B. Cooperative Sensing and Access Optimization
We discuss optimization of cooperative sensing and access
parameters to maximize the normalized throughput under
sensing constraints for PUs. In particular, the throughput
maximization problem can be stated as follows:
max
τij ,W
NT (τ, W) (15)
s.t. Pj
d εj
, τj
, aj ≥ Pj
d, j ∈ [1, M] (16)
0 τij
≤ T, 0 W ≤ Wmax, (17)
where Pj
d is the detection probability for channel j at the AP,
Wmax is the maximum contention window and recall that T
is the cycle interval.
Algorithm 1 SENSING AND ACCESS OPTIMIZATION
1: Assume we have the sets of all SU i, {Si}. Initialize τij
,
j ∈ Si.
2: For each integer value of W ∈ [1, Wmax], ﬁnd ¯τij
as
3: for i = 1 to N do
4: Fix all τi1j
, i1 = i.
5: Find optimal ¯τij
as ¯τij
= argmax
0τij ≤T
NT τij
, W .
6: end for
7: The ﬁnal solution ¯W, ¯τij
is determined as ¯W, ¯τij
=
argmax
W,¯τij
NT ¯τij
, W .
We propose a low-complexity algorithm (Alg. 1) to ﬁnd an
efﬁcient solution for the optimization problem (15, 16, 17).
In particular, for each potential value of W ∈ [1, Wmax], we
search for the best τij
to maximize the total throughput. This is
done by a sequential search technique. Then, the ﬁnal solution
is determined by the best combination of τij
, W for different
values of W. Numerical results reveal that Alg. 1 can always
ﬁnd the optimal solution of the underlying problem.
C. Channel Assignment for Throughput Maximization
So far we have assumed a ﬁxed channel assignment based
on which SUs perform sensing. In this section, we attempt to
determine an efﬁcient channel assignment solution by solving
the following problem
max
{Si}
NT ¯τij
, ¯W, {Si} (18)
Algorithm 2 CHANNEL ASSIGNMENT ALGORITHM
1: Run Alg. 1 for temporary assignments Si = S, i ∈ [1, N]
to get ¯W, ¯τij
. Employ Hungarian algorithm [14] to
determine the ﬁrst channel assignment for each SU so
that each channel is assigned to exactly one SU where the
cost of assigning channel j to SU i is ¯τij
. This results in
initial channel assignment sets {Si} for different SU i.
2: continue := 1, k := 1.
3: while continue = 1 do
4: Calculate the normalized throughput with optimized
parameter setting by using Alg. 1 as NT th =
NT ¯τij
, ¯W, Si .
5: Each SU i calculates the increase of throughput if it
is assigned one further potential channel j as ΔTij =
NT ¯τij
, ¯W, S1
i − NT th where S1
i = Si ∪ j and
¯τij
, ¯W are determined by using Alg. 1 for assignment
sets S1
i and Sl, l = i.
6: Find the “best” assignment (¯i, ¯j) as (¯i, ¯j) =
argmax
i,j∈SSi
ΔTij.
7: if ΔT¯i¯j δ then
8: Assign channel ¯j to SU ¯i (Si = S1
i ).
9: k = k + 1.
10: else
11: continue := 0
12: end if
13: end while
14: if k 1 then
15: Return to step 2.
16: else
17: STOP Alg.
18: end if
1) Brute-force Search Algorithm: Since the possible num-
ber of channel assignments is ﬁnite, we can employ the brute-
force search to determine the optimal channel assignment
solution and its protocol parameters. This can be done by
determining the best conﬁguration parameters under each
channel assignment (i.e., using Alg. 1) then comparing the
throughput achieved by different channel assignments to ﬁnd
the best one.
We now quantify the complexity of this optimal brute-force
search algorithm. The number of possible assignments is equal
to the following: How many ways are there to ﬁll 1/0 to the
elements of an NxM matrix. It can verify that the number
of ways is 2MN
. Therefore, the complexity of the optimal
brute-force search algorithm is O 2MN
. Moreover, for each
case, we must run Alg. 1 to determine the sensing and access
parameters.
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